On the size of the singular set of minimizing harmonic maps

We consider minimizing harmonic maps $u$ from $\mathrm{\Omega }\subset {\mathbb{R}}^n$ into a closed Riemannian manifold $\mathcal{N}$ and prove: (1) an extension to $n\ge 4$ of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$ is finite, we have $${\mathcal{H}}^{n-3}(\text{sing }u)\le C{\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }{\mathrm{\nabla }}_{T}u{\mathrm{\mid }}^{n-1}d{\mathcal{H}}^{n-1};$$ (2) an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is $\mathcal{N}={\mathbb{S}}^2$ we obtain that the singular set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary data. In dimension $n=3$ both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$ with $s\in (\frac{1}{2},1]$ and $p\in [2,\mathrm{\infty })$ satisfying $sp \geq 2$. We also discuss sharpness.